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On time-harmonic Maxwell equations with nonhomogeneous conductivities: Solvability and FE-approximation. (English) Zbl 0696.65085
The authors analyze the solvability of the 3D problem: \(rot H=E\) in \(\Omega,\) \(rot E=-i^{\omega \mu}H\) in \(\Omega\) and \(nx E=nx \tilde E\) on \(\partial\Omega,\) where E, H are complex-valued vector functions independent of time, \(\epsilon\) is the electric dielectricity, \(\mu\) the magnetic permeability, \(\sigma\) is the \(3\times 3\) matrix of electric conductivities, \(\tilde E\) a given vector function. An existence theorem for the solution is proved, and an error estimation for a finite element approximation of the 3D problem is presented. Some particular results for the 2D problem are obtained. Two numerical experiments are presented.
Reviewer: I.Grosu

65Z05 Applications to the sciences
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
78A25 Electromagnetic theory (general)
35Q99 Partial differential equations of mathematical physics and other areas of application
Full Text: EuDML
[1] V. Bezvoda K. Segeth: Mathematical modeling in electromagnetic prospecting methods. Charles Univ., Prague, 1982.
[2] E. B. Byhovskiy: Solution of a mixed problem for the system of Maxwell equations in case of ideally conductive boundary. Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 12 (1957), 50-66.
[3] V. Červ K. Segeth: A comparison of the accuracy of the finite-difference solution to boundary-value problems for the Helmholtz equation obtained by direct and iterative methods. Apl. Mat. 27 (1982), 375-390. · Zbl 0511.65074
[4] P. G. Ciarlet: The finite element method for elliptic problems. North-Holland, Amsterdam, New York, Oxford, 1978. · Zbl 0383.65058
[5] D. Colton L. Päivärinta: Far field patterns and the inverse scattering problem for electromagnetic waves in an inhomogeneous medium. Math. Proc. Cambridge Philos. Soc. 103 (1988), 561-575. · Zbl 0669.35109
[6] G. Duvaut J. L. Lions: Inequalities in mechanics and physics. Springer-Verlag, Berlin, 1976. · Zbl 0331.35002
[7] V. Girault P. A. Raviart: Finite element methods for Navier-Stokes equations. Springer- Verlag, Berlin, Heidelberg, New York, Tokyo, 1986. · Zbl 0413.65081
[8] P. Grisvard: Boundary value problems in nonsmooth domains. Lecture Notes 19, Univ. of Maryland, Dep. of Math., College Park, 1980.
[9] G. Heindl: Interpolation and approximation by piecewise quadratic \(C^1\)-functions of two variables. in Multivariate approximation theory by W. Schempp and K. Zeller. ISNM vol. 51, Birkhäuser, Basel, 1979, pp. 146-161.
[10] E. Hewitt K. Stromberg: Real and abstract analysis. Springer-Verlag, New York, Heidelberg, Berlin, 1975.
[11] J. Kadlec: On the regularity of the solution of the Poisson problem on a domain with boundary locally similar to the boundary of convex open set. Czechoslovak Math. J. 14 (1964), 386-393. · Zbl 0166.37703
[12] F. Kikuchi: An isomorphic property of two Hubert spaces appearing in electromagnetism. Japan J. Appl. Math. 3 (1986), 53-58. · Zbl 0613.46040
[13] M. Křížek P. Neittaanmäki: Solvability of a first order system in three-dimensional nonsmooth domains. Apl. Mat. 30 (1985), 307-315. · Zbl 0593.35073
[14] S. Mareš, al.: Introduction to applied geophysics. (Czech). SNTL, Prague, 1979.
[15] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. · Zbl 1225.35003
[16] P. Neittaanmäki R. Picard: Error estimates for the finite element approximation to a Maxwell-type boundary value problem. Numer. Funct. Anal. Optim. 2 (1980), 267-285. · Zbl 0469.65079
[17] J. Saranen: On an inequality of Friedrichs. Math. Scand. 51 (1982), 310-322. · Zbl 0524.35100
[18] R. S. Varga: Matrix iterative analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1962. · Zbl 0133.08602
[19] V. V. Voevodin, Ju. A. Kuzněcov: Matrices and computation. (Russian). Nauka, Moscow, 1984.
[20] Ch. Weber: A local compactness theorem for Maxwell’s equations. Math. Methods Appl. Sci. 2 (1980), 12-25. · Zbl 0432.35032
[21] Ch. Weber: Regularity theorems for Maxwell’s equations. Math. Methods Appl. Sci. 3 (1981), 523-536. · Zbl 0477.35020
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